The occurrence of nondescent directions in successive quadratic programming is studied. It is shown that simple chemical process examples can be constructed that exhibit nondescent as a consequence of the projected indefiniteness of the Hessian matrix of the Lagrangian function. Moreover, in situations where multiple Kuhn-Tucker points for the quadratic programming subproblems exist, the global optimum need not necessary provide a direction of descent. Thus search for a global solution is unjustified. To circumvent these difficulties, a linear programming-based trust region method is proposed to guarantee descent for any arbitrary merit function, provided such a direction exists. Geometric illustrations are used to elucidate the main ideas.